Mathematics for the interested outsider

Enriched Naturality Revisited

Let’s look back at the enriched versions of representable functors. If we fix an object we have a -natural transformation . This corresponds under the closure adjunction to . There’s a similar transformation for composition on the other side.

Now remember that the closest thing we have to a “morphism” in an enriched category is an element of the underlying set of a hom-object. That is, we can talk about an arrow . We often abuse the language and say that this is a morphism from to , which in fact it’s a morphism in the underlying category.

Now, even though this isn’t really a morphism in our enriched category, we can still come up with a morphism sensibly called . Here’s how it goes:

We start with and use the left unit isomotphism to move to .

We now hit with our morphism to land in .

Finally, we compose to end up in .

We can similarly take and construct a morphism .

These composites should look familiar from the definition of enriched naturality for a transformation . In fact, we have a more compact diagram to replace that big hexagon:
Notice here that the right and bottom arrows in this square expand out to become the top and bottom of a hexagon, and we slip the functors and into place.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.